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Lecture 4: Frequency domain

representation, DTFT, IDTFT, DFT, IDFT

Instructor:

Dr. Gleb V. Tcheslavski

Contact:gleb@ee.lamar.edu

Office Hours: Room 2030

Class web site:

http://ee.lamar.edu/gleb/dsp/index.htm

mailto:gleb@ee.lamar.eduhttp://ee.lamar.edu/gleb/dsp/index.htmhttp://ee.lamar.edu/gleb/dsp/index.htmmailto:gleb@ee.lamar.edu

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Some history

Jean Baptiste Joseph Fourier was born

in France in 1768. He attended the Ecole Royale

Militaire and in 1790 became a teacher there.

Fourier continued his studies at the Ecole

Normale in Paris, having as his teachersLagrange, Laplace, and Monge. Later on, he,

together with Monge and Malus, joined Napoleon

as scientific advisors to his expedition to Egypt

where Fourier established the Cairo Institute.

In 1822 Fourier has published his most

famous work: The Analytical Theory of Heat.Fourier showed how the conduction of heat in

solid bodies may be analyzed in terms of infinite

mathematical series now called by his name, the

Fourier series.

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Frequency domain representation

0cos( ) a (co)sinusoidal inputnLet x A n = +

frequency

0 00 0( ) ( )

00 00

j n jj njn j n

n

jA Ae e e eA Ax e e + +

+= + =

complex exponent of0

complex exponent of -0

A sinusoidal signal is represented by TWO complex exponents of

opposite frequencies in the frequency domain.

(4.3.1)

(4.3.2)

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Frequency domain representation (cont)

( )0 0 0 00( )j n j n k j k j nn k kk

n

k

j

nLet x e h e hy H xe ee = = =

@

( ) ( )( )

: phase

jj H e

j j k j

k

kmagnitude

The frequency respon H e h e H e ese

= =00000

00000

Iff it exists!

( ) ( )

( )

( )

( )

( )

( ) ( )

( ) ( )

* *

*

:

j j

j n j n j n

n n n

n n n

j H e j H e

j j

j j

j j

j j

real H e H For a system h e h e h e

H e e H e

e

H e H e

H e H e

e

= = =

=

=

=

=

has a period of 2( )jH e

(4.4.1)

(4.4.2)

(4.4.3)

(4.4.4)

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Frequency domain representation (cont 2)

For an arbitrary

real LTI system:

0 .00 .00 .00 .00 0 .00 .00 .00 .00 0-000

-00

0

00

000

Normalized Frequency ( rad/sample)

Phase(degree

s)

0 .00 .00 .00 .00 0 .00 .00 .00 .00 0-00

-00

-0

0

0

00

Normalized Frequency ( rad/sample)

Magnitude(dB)

Symmetric with

respect to

Anti-symmetric

with respect to

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Frequency domain representation (cont 3)

Combining(4.3.2)and(4.4.4) back to our sinusoid!

( ) ( ) { }

( )( )

( )( )

( )( ) ( )

0 0 0 0

0 0

0 0 0 0

0 00 0

0

Stady-state response:0 0

0 0

0

j j

j j

j j n j j nj j

n

j H e j H ej j n j j nj j

j n H e j n H ej

A Ay e H e e e H e e real system

A A

e H e e e e H e e e

AH e e e

+ + + +

= + =

= +

= +

LTI filtering: ( ) ( )

( )

0 0

0cos

j j

ny A H e n H e = + +

due to the

input

change due to the system

same as

the input

from the

input

phase change due to the system

Via design, we manipulate H(ej), therefore, hn, and, finally, manipulate the

coefficients in the Linear Constant Coefficient Difference Equation (LCCDE)

(4.6.1)

(4.6.2)

(4.6.3)

(4.6.4)

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8/3/2019 Lecture 04 - Frequency

7/38ELEN 5346/4304 DSP and Filter Design Fall 2008

7

Frequency domain representation (cont 4)

LCCDE:

0

0

,

, 0

0

j n

j nnn

tot n i i

i

x e ny c ke

initial conditions

= = +

if the system is BIBO( ) , the frequency response existsj j n

n n

n n

H e h e h

= <

for large enough n:0

,

j n

tot ny ke

( )0 0 00 0

0

( )n

j n k j k j n

n k n k k

k k

j

n

j ny h e u e ee Hh e

=

= = =

0000000

(4.7.1)

(4.7.4)

(4.7.2)

(4.7.3)

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8/3/2019 Lecture 04 - Frequency

8/38ELEN 5346/4304 DSP and Filter Design Fall 2008

8

Frequency domain representation (cont 5)

for an LTI:0

, ,

j n

n n n tr n ss nx e u y y y= = +

We dont need systems of order higher than 2: can always make cascades.

0 0

,

( ) ,j j n

ss n

y H e e exists only for LTI BIBO systems =

for a real, LTI, BIBO system:

0 0

,0 0cos( ) ( ) cos( ( )j j

n ss nx A n y A H e n H e = + = + +

effects of filtering

We cannot observe ANY frequency components in the output that are not

present in the input (in steady state). We may see less when ( ) 0rjH e =

(4.8.1)

(4.8.2)

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8/3/2019 Lecture 04 - Frequency

9/38ELEN 5346/4304 DSP and Filter Design Fall 2008

9

Frequency domain representation (cont 6)

In continuous time:

{we want}t t t t t x s n y c s distortion less transmission= + ; = signal noise const delay

( )( ) ( ) ( ) ( ) ( )j t j t j jt tY y e dt c s e d t e ce S H S

= = = =

We need a constant magnitude and linear phase for the frequencies of interest.

Ideal filters:

0( )j

H e

0( )jH e

LPF

0( )j

H e

HPF

0( )j

H e

BPF

0( )j

H e

BSF

Ideal filters are non-realizable!

(4.9.1)

(4.9.2)

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8/3/2019 Lecture 04 - Frequency

10/38ELEN 5346/4304 DSP and Filter Design Fall 2008

10

CTFT and ICTFT

(4.10.1)

(4.10.2)

( ) ( )

0( ) ( )

0

j t

j t

X x t e dt

x t X e d

=

=

CTFT:

ICTFT:

Examples:

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8/3/2019 Lecture 04 - Frequency

11/38ELEN 5346/4304 DSP and Filter Design Fall 2008

11

DTFT

( ) , ,j j nnn

a complex funX e x ction periodic continuous one

=

0= if exists

(4.11.1)

Whats about convergence???

1. Absolute convergence:

( )0 i.e. converges absolutely or uniformlyj

n n

n

if x l x X e , < ,

( ) ( )

.

( ) lim 0k

j j n

k n

n k

abs er

j j

kk

ror

Le X et X e x e X e

=

= =000000000

(4.11.4)

(4.11.5)

( ) ( ) ( )( ) ( ) ( ) ( )( )

( ) ( ) ( )0

0 0

cos , sinj j j j j jre im

j j j

re im

X e X e X e X e X e X e

X e X e X e

= =

= +

(4.11.2)

(4.11.3)

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8/3/2019 Lecture 04 - Frequency

12/38ELEN 5346/4304 DSP and Filter Design Fall 2008

12

DTFT (cont)

0

0 0

0

. , :cos( )

n

n

n

n

u

a are not l In fact they are not l xA n

< +

must be

2. Mean-square convergence:

( ) ( )lim 0

energy of the

j j

err

k

r

k

o

X e X e

=000000000

The total energy of the error must approach zero, not an error itself!

(4.12.1)

Absolutely summable sequences always have finite energy. However,

finite energy sequences are not necessary absolutely summable.

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8/3/2019 Lecture 04 - Frequency

13/38ELEN 5346/4304 DSP and Filter Design Fall 2008

13

IDTFT

( )0

,0j n

j p

n

n

Let p is a period X e x e

=

= =

00( ) ( )

0

j j n

p

j j n

nx XX e e dp

e e d

== IDTFT:

(4.13.1)

(4.13.2)

Combining (4.11.1) and (4.12.2)0

0

j l j n

n l

l

x x e e d

=

=

[ ]( ) sin ( )0( )0

j n l

l l l n l nl l

n l

x e d x x xn l

= =

= = =

(4.13.3)

(4.13.4)

( )jX e shows wherexn lives in the frequency domain.

14

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8/3/2019 Lecture 04 - Frequency

14/38ELEN 5346/4304 DSP and Filter Design Fall 2008

14

Back to ideal filters

Ideal LPF:

0( )jH e

20

1

c

( ),0 0

,0

cj

LP

c

H e

= <

8/3/2019 Lecture 04 - Frequency

15/38ELEN 5346/4304 DSP and Filter Design Fall 2008

15

DTFT properties

( ) ( )

( )

( )

0

0

0 0( )

:

:

:

:

j j

n nDTFT

j n j

n nDTFT

j n j

nDTFT

nD

Linearity ag bx aG e bX e

Shift in time g e G e

Shift in frequency e g G e

Differentiation n g

+ +

( )

( ) ( )

( ) ( )

( ) ( )

( )

* *

:

0:0

0' :

0

j

TFT

j j

n nDTFT

j jn n

DTFT

j j

n n

n

dG ej

d

Linear convolution g x G e X e

Periodic convolution g w G e W e d

Parseval s theorem g w G e W e d

=

=

(4.15.1)

(4.15.2)

(4.15.3)

(4.15.4)

(4.15.5)

(4.15.6)

continuous, periodic functions

(4.15.6)

16

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8/3/2019 Lecture 04 - Frequency

16/38ELEN 5346/4304 DSP and Filter Design Fall 2008

16

DTFTs of commonly used sequences

17

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8/3/2019 Lecture 04 - Frequency

17/38ELEN 5346/4304 DSP and Filter Design Fall 2008

17

DTFT examples

( )0 0( )0 0j n j

nDTFT

k

Let x e X e k = = +

( )0 ( )0 0 0j

nDTFT

k

x X e k = = = +0

( ) ( ) ( )00

( )0 0 0 00

j j j

n n n jDTFTl

u u U e e U e c l e

= = = + + of DC value ofun0

( )00

n jDTFT

l

u le

+ +

( )*

* *j n j n j

n n

n n

h e h e H e = =

( ) ( )

( ) ( )

( )

( )

0 0

0 0

0

0

j n j j n j n j j n

n n n n

n n n

j j

h g e H e e g e d H e g e d

H e G e d

= =

=

(4.17.1)

(4.17.2)

(4.17.3)

(4.17.4)

(4.17.5)

(4.17.6)

18

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8/3/2019 Lecture 04 - Frequency

18/38ELEN 5346/4304 DSP and Filter Design Fall 2008

18

DTFT examples (cont)

We can re-work the Parsevals theorem (4.15.6) as follows:

( ) ( ) ( )(

0

)

0 * 0

0

0

0j

hS

j

n n

j

n

j

e

Energy HE h de ed HH e

: == = 00000

energy density (spectrum)

Autocorrelation function:

( ) ( ) ( )*( ),* * * j j

n n l n l n l

j

gg l gDTFT

l

n n

g g g g g gr G e G Se e = = =

(4.18.1)

(4.18.2)

19

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8/3/2019 Lecture 04 - Frequency

19/38ELEN 5346/4304 DSP and Filter Design Fall 2008

19

DTFT examples (cont 2)

One obvious problem with DTFT is that we can never compute it sincexn

needs to be known everywhere! which is impossible!

Therefore, DTFT is not practical to compute.

Often, a finite dimension LTI system is described by LCCDE:

( ) ( )

( )( )

( )

0 0 0

0

0

0

N M N Mj i j j m j

i n i m n m i mDTFT

i m i m

j

j

Mj m

mjm

Nj i

i

i

b eH e

a

a y b x a e Y e b e X e

Y e

X ee

=

= = =

=

=

=

=

=

=

(4.19.2)

practical (finite

dimensions)

Prediction of steady-state behavior of LCCDE

(4.19.1)

20

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8/3/2019 Lecture 04 - Frequency

20/38ELEN 5346/4304 DSP and Filter Design Fall 2008

20

How to measure frequency response

of an actual (unknown) filter?

1. Perform two I/O experiments:

( ) ( )( )

( ) ( )( )

0 0

0 0

( ) ( ) ( )

,0 0

( ) ( ) ( )

,0 0

) cos( ) cos0

) sin( ) sin0

j jc c c

n n tr nLTI

j js s s

n n tr n

LTI

x A n y A H e n H e y

x A n y A H e n H e y

= + = + + +

= + = + + +

2. Analyze these measurements and form:

( )( )

( ) ( )

0

00

0

0

0

( )( ) ( )

( ) (

,

)

,

j

j

j nc s

n n n

j n H ejc s

j H e

n

jntr n

n

n n tr n

x x jx Ae

y y jy A H e e y

Finallyy

H e e zx

+

+ +

= + =

= + = +

= +%

%

%

%

%

Thats a good way to measure/estimate a frequency response for every .

(4.20.1)

(4.20.2)

(4.20.3)

(4.20.4)

(4.20.5)

21

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8/3/2019 Lecture 04 - Frequency

21/38ELEN 5346/4304 DSP and Filter Design Fall 2008

21

DFT and IDFT

Consider an N-sequencexn (at most Nnon-zero values for0 n N-1)

( ) ( )0

0

0

, , ,...,00 0k

Nj n j

n kk

n

n

j j

n N

n x e X eX e x e kX N

=

=

= = == = uniformly spaced frequency samples

(4.21.1)

00

0

, , ,...,00 0N j kn

Nk n

n

X x e k N

=

= = (4.21.2)DFT:Finite sum! Therefore, its computable.

0

Using notation:j

NNW e

=

(4.21.1) can be rewritten as:0

0

Nkn

k n N

n

X x W

=

= 0

0

: , ,.00, ..,0

0

Nkn

n k N

n

IDFT n Nx X WN

=

= =

(4.21.3)

(4.21.4)

(4.21.5)

Btw, DFT is a sampled version of DTFT.

22

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8/3/2019 Lecture 04 - Frequency

22/38ELEN 5346/4304 DSP and Filter Design Fall 2008

22

DFT and IDFT (cont)

Let us verify (4.21.5). We multiply both sides by l nNW

0 0 0 0 0

( )

0 0 0 0 0

0 0N N N N N

l n kn l n k l n

n N k N N k N

n n k n k

x W X W W X WN N

= = = = =

= =

0 0 0

( )

0 0 0

0N N N

l n k l n

n N k N

n k n

x W X WN

= = =

=

0

( )

0

, ,

,0

Nk l n

Nn

N for k l rN r is Integer

Since W otherwise

=

=

= 0

0

Nl n

n N l

n

x W X

=

=

(4.22.1)

(4.22.2)

(4.22.3)

(4.22.4)

23

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8/3/2019 Lecture 04 - Frequency

23/38ELEN 5346/4304 DSP and Filter Design Fall 2008

23

FFT

In the matrix form: N D TD x FX =

where: 0 0 0[ ... ]T

Nx x x x time domain signal=

0 0 0[ ... ]TNX X X X N DFT samples

=

0 0 0

( )0 0 0 0

( ) ( ) ( )0 0 0 0 0

0 0 0 0

0

0

0

N

N N N

N

N N N N

N N N N

N N N

W W W

D W DFT mW W

W W W

atrix

=

L

L

L

M M M O M

L

0

Nx D X IDFT =

( )0 0 0

( )0 0 0 0 0

( ) ( ) ( ) ( )0 0 0 0 0

0 0 0 0

00

0

0

N

N N N

N

N N N N

N N N N

N N N

W W W

D W W WN

W W W

IDFT matrix

=

L

L

L

M M M O M

L *0 0N ND D

N

=

(4.23.1)

(4.23.2)

(4.23.3)

(4.23.4)

(4.23.5)

(4.23.6)

(4.23.7)

This is actually FFT

24

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8/3/2019 Lecture 04 - Frequency

24/38ELEN 5346/4304 DSP and Filter Design Fall 2008

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Relation between DTFT and DFT

1. Sampling of DTFT

( )jnDTFT

Let x X e

( ) ( ){ }, , ,... point DFT0 00 0 kjj k kk

Sample X e at k N N frequency samples X e Y N N

= = =

, , ,...,00 0k nIDFTY y n N =

( )0

k

kj

j klNk l N

l

Y X e X e x W

=

= = =

0 0 0

( )

0 0 0

0 0 0N N N

kn kl kn k n l

n k N l N N l N

k k l l k

y Y W x W W x W

N N N

= = = = =

= = =

, , ,...,00 0n n Nm

mConsidering y x n N

=+ = =( . .000 )0

(4.24.1)

(4.24.2)

(4.24.3)

yn is an infinite sum of shifted replicas ofxn. Iffxn is a length Msequence (M N)

than yn = xn. Otherwise, time-domain aliasing xncannot be recovered!

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